Difference continous - discrete symmetry I am trying to understand the difference between the  two types of symmetries.Wiki
Wikipedia says that


*

*Translation in time : $t \rightarrow t + a$
is a $\textbf{continuous}$ symmetry, for any real $t,a$
but

*Time reversal: $t \rightarrow -t$ is a $\textbf{discrete}$  symmetry.


But if we choose $a=- 2t$ we get the same transformation - can somebody explain to me why this is no contradiction?
 A: When we say a symmetry is continuous, it is shorthand for saying that the group of symmetries is continuous.
For time translations, the group consists of all translations for any a, where $a$ is a continuous parameter. There are an infinite number of these group elements, and there are elements of the group which are infinitely close together.
For time reversal, there are only two group elements and they are not close together in any sense.
A: In a nutshell, $a$ is an arbitrary but fixed 1-parameter, while $t$ is a running time coordinate. One cannot consistently put a fixed parameter equal to a running coordinate. Phrased differently, $a$ is here not allowed to depend on $t$. 
In particular, the 1-parameter family $(\mathbb{R}\ni t\mapsto t+a\in\mathbb{R})_{a\in\mathbb{R}}$ of time translation maps is a continuous deformation of the identity map $\mathbb{R}\ni t\mapsto t\in\mathbb{R}$.
In contrast, the time reversal map $\mathbb{R}\ni t\mapsto -t\in\mathbb{R}$ is a discrete deformation of the identity map $\mathbb{R}\ni t\mapsto t\in\mathbb{R}$.
